Model II: Medical Resource Is Limited
6.2 Problem Description
As mentioned before, we have divided the entire emergency rescue process into three stages in Liu and Zhao [5], and this study focuses on the optimization of the emergency logistics network in the third rescue stage. In such stage, situation of the epidemic diffusion tends to be stable and the spread of the epidemic goes to under control. Thus, optimization goal in such stage is to construct an integrated, dynamic and multi-level emergency logistics network, which includes the national strategic storages, the urban health departments and the epidemic areas. The research idea of the third emergency rescue stage is shown in Fig.6.1.
National strategic storage
01 2 3 4 5
n
Urban health department Epidemic area
Time duration Decision -making cycle (a)
(b)
(c)
Distribution arc
a, b, c
Fig. 6.1 Research idea of the third emergency rescue stage
As Fig.6.1shows, the entire rescue process in the third emergency rescue stage is decomposed into several mutually correlated sub-problems (i.e.ndecision-making cycles). To each decision-making cycle, there exist two sub-problems. In the upper level, we consider the problem how to replenish emergency resources to the urban health departments. Besides, we adjust the replenishment arcs by a heuristic algo- rithm, and construct a mixed-collaborative delivery system. Thus, the total rescue cost of the upper level sub-problem would be minimized. In the lower level, we present the problem how to allocate emergency resources to the infected areas. We propose a forecasting model for the time-varying demand in the epidemic areas based on the epidemic diffusion rule. Such two phases are executed iteratively. Besides, at the end of each rescue cycle, effect of emergency resources allocated is analyzed and the number of infected people is updated. Such a sequential operational routine is continued until the bio-virus diffusion is under control.
It is worth mentioning that the optimal result of the upper level sub-problem affects the result of the lower level sub-problem, directly; on the other side, the optimal result of the lower level sub-problem will affect the result of the upper level sub- problem in the next emergency rescue cycle. Therefore, this is different to the bi-level programming method. In what follows, we will present the SEIR epidemic diffusion model and the forecasting models for the time-varying demand and inventory.
6.2.1 SEIR Epidemic Diffusion Model
Since most epidemics divide people into four classes: the susceptible people (S), the people during the incubation period (E), the infected people (I), and the recovered people (R). Thus, as Fig.6.2shows, without consideration of the population migra- tion, and the natural birth and death rate of the population, we can use a SEIR model based on small-world network to describe the developing epidemic process.
Therefore, the following SEIR model [6] is adopted in this study.
⎧⎪
⎪⎨
⎪⎪
⎩
d S
dt = −βkS(t)I(t)
d E
dt =βkS(t)I(t)−βkS(t−τ)I(t−τ)
d I
dt =βkS(t−τ)I(t−τ)−(d+δ)I(t)
d R
dt =δI(t)
(6.1)
In such epidemic diffusion model, the time-based parameters S(t), E(t), I(t) andR(t), represent the number of susceptible people, the number of people during the incubation period, the number of infected people, and the number of recovered people, respectively. Other parameters include:kis the average degree distribution of the small-world network;β is the propagation coefficient of the bio-virus (small- pox);δis the recovered rate of the infected people;dis the death rate caused by the disease;τ stands for the incubation period. Furthermore,k, β, δ,d, τ >0.
From the Eq. (6.1), we can see thatI(t), which denotes the number of infected people, can be calculated by solving the ordinary differential equations when the initial values ofS(t),E(t),I(t)andR(t)are given. Actually, this parameter is one of the most important concerns during the emergency rescue process, and it is desired thatI(t)stays at a value as low as possible, which implies that the situation is stable and the spread of the epidemic is under control. Wang et al. [4] propose that the change of I(t)mainly depends on the population of the recovered people and the onset people at the end of the incubation period. And thus, we should improve the recovered rateδ and reduce the propagation coefficientβ, thereby decreasing the value ofI(t)effectively.
β<k>S(t-τ)I(t-τ) δI
d1I β<k>SI
S E I R
Fig. 6.2 SEIR epidemic diffusion model
6.2.2 Forecasting Model for the Time-Varying Demand
As mentioned before, for both upper and lower sub-problems are existed in each emergency rescue cycle, thus, the time-varying demand in each sub-problem should be the forecasted respectively.
(1) Forecasting model for the time-varying demand in the epidemic area As introduced in Sect.6.1, the demand information is quite limited and varies rapidly with time when suffered from a bioterror attack. Thus, it is often difficult to predict the actual demand based on historical data. Xu et al. [7] propose that demand forecasting after a disaster is especially important in emergency management, and present an EMD-ARIMA (empirical mode decomposition and autoregressive integrated moving average) forecasting methodology to predict the agricultural products demand after the 2008 Chinese winter storms. Other related works can be found in [8,9]. Note that emergency demand in the previous literature has always been formulated as a stochastic or deterministic variable, while the effectiveness that emergency resource allocated in the early rescue cycle will affect the demand in the later rescue cycle has not been considered. Based on the previous works ([5]), the following forecasting model for the time-varying demand in the epidemic area is adopted in this study.
dt∗ =a I(t), t ∈0,1,2, . . . ,n (6.2) ηt =(dt∗+1−dt∗)
dt∗, t∈0,1,2, . . . ,n−1 (6.3)
When t =0, d0=a I(0) (6.4)
When t =1, d1=(1+η0)
1− θ
d0 (6.5)
When t =2, d2=(1+η1)
1− θ
d1=(1+η0)
1+η1)(1− θ
2
d0 (6.6) . . .
When t =n, dn =
n−1 i=0
(1+ηi)
1− θ
n
d0 (6.7)
Herein,n−1
i=0(1+ηi)=(1+η0)(1+η1) . . . (1+ηn−1). Equation (6.2) is the traditional forecasting model for the time-varying demand.dt∗means demand of the emergency resources in the epidemic area at timet,t ∈ 0,1,2, . . . ,n.I(t)is the number of infected people in the epidemic area at time t.a is the proportionality coefficient. Equation (6.3) is used to calculate the linear scale factor of the change in demand for each rescue cycle. Furthermore,ηt ≤0.d0in the Eq. (6.4) is the initial
demand of emergency resources in the epidemic area, andI(0)represents the initial number of infected people in the epidemic area.d1,d2, . . . ,dn in Eqs. (6.5)–(6.7) represent the demand of emergency resources in emergency rescue cycle 1,2, . . . ,n.
Other parameter include:θis the effective rescue rate in each cycle;is the treatment cycle for each infected person. To facilitate the calculation process in the following sections, we assume thatis an integral multiple of the rescue cycle.
According to the above recursion formulas, the change of emergency demand mainly depends on these two important parameters. Thus, in the context of emergency rescue, there should be enough emergency resources to cure the infected people, so that the effective rescue rateθ can be improved and the treatment cycleΓ can be reduced, thereby, decreasing the total emergency rescue cost.
(2) Forecasting model for the time-varying demand in urban health depart- ment
As introduced before, in the upper level sub-problem, we consider the problem how to replenish emergency resources to the urban health departments. Thus, the urban health departments, which are the emergency suppliers in the lower level sub- problem, have been changed to be the demand nodes in the upper level replenishment network. Note that time-varying demand in the urban health department mainly depends on the unsatisfied capacity. Hence, to facilitate the calculation process in the following sections, we assume that the initial inventory in each urban health department is equal to zero. Besides, we suppose that capacity of each urban health department is equal toVcap. Supposing thatdtvrepresents the demand of emergency resources in urban health department at rescue cyclet,Ptrepresents the total supply of the emergency resources in urban health department at rescue cyclet(Such value is obtained by solving the lower level sub-problem in the previous rescue cycle).
Thus, the forecasting model for time-varying demand in urban health department can be formulated as follows.
dtv=
Vcap, t =0
Pt−1,t =1,2, . . . ,n (6.8)
6.2.2.1 Forecasting Model for the Time-Varying Inventory
As mentioned in Sect.6.1, the focus of this study is placed on replenishing emergency resources to the urban health departments and distributing them to the epidemic areas, simultaneously. Thus, the urban health departments play the role of the link in the multi-level emergency logistics network. Intuitively, inventory of the emergency resources in the urban health department should also be changed as time goes by.
Supposing that Vt is the inventory of the emergency resources in the urban health department at rescue cyclet, and we can get the following equation.
Vt=
0, t =0
Vcap−Pt−1,t =1,2, . . . ,n (6.9)